Chapter 6 Reference axis-Centroid by inspection

NOTE: SOLUTIONS FOR RADIUS OF GYRATION PROBLEMS 6-57 TO 6-74 ARE

INCLUDED WITHIN PROBLEMS 6-2 TO 6-29

FIGURE P6-1 Units: Inches NOTE: L-shape

Part Area y Ay IcdAd

2

Ic + Ad2

1-Vertical 0.5000 1.0000 0.5000 0.1667 0.3365 0.0566 0.2233

FIGURE P6-3 Units: Inches NOTE: 8×8 rectangle with inner rectangle 4×6 rectangle removed

Part Area y Ay IcdAd

2

Ic + Ad2

1-total 8x8 48 0.00 0.00 256.00 0.00 0.00 256.0

2-Inner 4x6 –24 0.00 0.00 -72.00 0.00 0.00 –72.0

Total area = 24.00 Sum Ay = 0.0000 Total I = 184.0 in4

Y = 0.0000 Inches Reference axis-Centroid

rx = (I/A)= 2.769 in Problem 6-58

FIGURE P6-4 Units: mm NOTE: Tee shape

Part Area y Ay IcdAd

2

Ic + Ad2

SPREADSHEET SOLUTIONS FOR PROBLEMS 6-1 THROUGH 6-74

FIGURE P6-6 Units: mm NOTE: U-shape;Two vert rectangles (10x30) combined

Part Area y Ay IcdAd

2

Ic + Ad2

1-Ver-20x30 600 15.0 9.000E+03 4.500E+04 2.5 3.750E+03 4.875E+04

FIGURE P6-7 Units: mm NOTE: Z–Shape;Vertical stem 40x5;2 flanges 15x5

Part Area y Ay IcdAd

2

Ic + Ad2

1-Vertical 5×40 200 20.0 4000 2.667E+04 0.00 0.000E+00 2.667E+04

2-Horiz-bot 5x15 75 2.5 188 1.563E+02 17.50 2.297E+04 2.313E+04

3-Horiz-top 5x15 75 37.5 2813 1.563E+02 17.50 2.297E+04 2.313E+04

Total area = 350 Sum Ay = 7000 Total I = 7.292E+04 mm4

Y = 20.00 mm Reference axis-Base

FIGURE P6-8 Units: mm NOTE: H-shape;Two vert rectangles (5x40) combined

FIGURE P6-9 Units: mm NOTE: 40x40 rectangle with 35x30 rectangle removed

Part Area y Ay IcdAd

2

Ic + Ad2

1-total 40×40 1600.0 0.00 0.00 213333 0.00 0.00 2.133E+05

2-Inner 35×30 -1050.0 0.00 0.00 -78750 0.00 0.00 –7.875E+04

Total area = 550.00 Sum Ay = 0.0000 Total I = 1.346E+05 mm4

Y = 0.0000 mm Reference axis-Centroid-by inspection

rx = (I/A)= 15.643 in Problem 6-63

FIGURE P6-10 Units: mm NOTE: 28x60 rectangle with 23x50 rectangle removed

FIGURE P6-11 Units: mm NOTE: Hat–section; 2 verticals combined (10x45)

Part Area y Ay IcdAd

2

Ic + Ad2

1-Vertical 10x45 450 22.5 10125 75938 0.69 2.140E+02 7.615E+04

2-Horiz-bot 5x30 150 2.5 375 312.50 19.31 5.593E+04 5.625E+04

FIGURE P6-12 Units: mm NOTE: 4 verticals (4x16) combined; top (4x50)

Part Area y Ay IcdAd

2

Ic + Ad2

FIGURE P6-13 Units: mm NOTE: Modified ‘Chair’ shape

Part Area y Ay IcdAd

2

Ic + Ad2

1 Hor 5x10 50 2.5 125.0 104.2 20.833 2.170E+04 2.181E+04

FIGURE P6-14 Units: Inches NOTE: Hat section with bottom plate

Part Area y Ay IcdAd

2

Ic + Ad2

1-Bot plate 0.520 0.100 0.052 0.0017 0.433 0.098 0.099

2-Bot flanges 0.120 0.250 0.030 0.0001 0.283 0.0096 0.0097

FIGURE P6-15 Units: Inches NOTE: Box with two vertical stems above

Part Area y Ay IcdAd

2

Ic + Ad2

1-Bot flange 0.100 0.05 0.0050 0.0000833 1.0176 0.10356 0.1036

FIGURE P6-16 Units: Inches NOTE: Rectangle with semicircle top

Part Area y Ay IcdAd

2

Ic + Ad2

1-Rectangle 1.1250 0.7500 0.84375 0.21094 0.1492 0.0250 0.2360

FIGURE P6-17 Units: mm NOTE: Vertical rectangle with top and bottom semicircles

Part Area y Ay IcdAd

2

Ic + Ad2

1-Rectangle 11400 0.0 0 3.430E+07 0.00 0.000E+00 3.430E+07

FIGURE P6-18 Units: mm NOTE: Rectangle; Triangle on top; Square – Removed

Part Area y Ay IcdAd

2

Ic + Ad2

1-Rectangle 1200 20.0 24000 1.600E+05 7.27 6.347E+04 2.235E+05

Y = 27.27 mm Reference axis–Base

FIGURE P6-19 Units: Inches NOTE: Composite shape with two areas removed

Part Area y Ay IcdAd

2

Ic + Ad2

1-Rect. 0.6x2.5 1.5000 1.250 1.8750 0.781250 0.319 0.153 0.9343

FIGURE P6-20 Units: Inches NOTE: Channel with horizontal web; tapered flanges down

Part Area y Ay IcdAd

2

Ic + Ad2

1-Vert Rect (2) 1.200 1.000 1.2000 0.4000000 0.2798 0.09397 0.4940

FIGURE P6-21 Units: mm NOTE: Hat section with round top

Part Area y Ay IcdAd

2

Ic + Ad2

1-Bot Flange (2) 72.00 1.500 108.0 54.00 13.143 12436.9 12491

FIGURE P6-22 Units: Inches NOTE: Two rectangles with semicircle top

Part Area y Ay IcdAd

2

Ic + Ad2

1-Rectangles 1.2500 0.6250 0.78125 0.16276 0.3906 0.1908 0.3535

FIGURE P6-23 Units: Inches Note: Wood I-shape from three boards;Two 2x4s; one 2x6

Part Area y Ay IcdAd

2

Ic + Ad2

1-Vert 2×6 8.25 0.00 0.00 20.797 0.00 0.00 20.797

FIGURE P6-24 Units: Inches NOTE: Box-wood boards; Large rectangle–inner rectangle removed

Part Area y Ay IcdAd

2

Ic + Ad2

1-Outside box 50.750 0.000 0.000 207.229 0.0000 0.0000 207.23

FIGURE P6-25 Units: Inches NOTE: Box-wood boards; Large rectangle–inner rectangle removed

Part Area y Ay IcdAd

2

Ic + Ad2

1-Outside box 108.0 0.000 0.000 182.250 0.0000 0.0000 182.25

FIGURE P6-26 Units: Inches NOTE: Tee made from three 2x12 wood boards

Part Area y Ay IcdAd

2

Ic + Ad2

1-Verticals (2) 33.8 5.625 189.844 355.957 2.125 152.402 508.36

FIGURE P6-27 Units: mm NOTE: Box–SI wood; Large rectangle-inner rectangle removed

Part Area y Ay IcdAd

2

Ic + Ad2

1-Outside box 29070 0.000 0.000 5.671E+07 0.0000 0.0000 5.671E+07

FIGURE P6-28 Units: mm NOTE: Tee made from three 45x290 SI wood boards

Part Area y Ay IcdAd

2

Ic + Ad2

1-Verticals (2) 26100 145.000 3.785E+06 355.957 55.83 1.829E+08 1.829E+08

FIGURE P6-29 Units: Inches NOTE: W14x43 beam with 0.5×8.0 in cover plates, top & bottom

Part Area y Ay IcdAd

2

Ic + Ad2

1-W14x43 12.60 0.00 0.00 428.00 0.00 0.00 428.00

FIGURE P6-30 Units: Inches NOTE: S12x50 with C12×25 on top flange

Part Area y Ay IcdAd

2

Ic + Ad2

1-S12x50 14.600 6.000 87.600 303.00 1.913 53.43 356.43

FIGURE P6-31 Units: Inches NOTE: Aluminum I-12x14.292 with 0.5x7 plate on top flange

Part Area y Ay IcdAd

2

Ic + Ad2

1-I-12x14.292 12.150 6.000 72.900 317.33 1.398 23.74 341.07

FIGURE P6-32 Units: Inches NOTE: Two aluminum C12x8.274 with top and bottom plates

Part Area y Ay IcdAd

2

Ic + Ad2

1-C12×8.274 7.036 0.000 0.000 159.7600 0.000 0.00 159.76

FIGURE P6-33 Units: Inches NOTE: I-shape fabricated from 4 2x2x3/8 angles + 3 plates

Part Area y Ay IcdAd

2

Ic + Ad2

1 Vert plate 3.00 0.000 0.00 9.00 0.000 0.000 9.000

2-Bot angles (2) 2.74 -2.368 -6.49 0.9520 2.368 15.364 16.316

FIGURE P6-34 Units: Inches NOTE: Fabricated box shape; two channels with two side plates

Part Area y Ay IcdAd

2

Ic + Ad2

1-Vert plates (2) 3.000 0.000 0.000 9.000 0.000 0.000 9.000

FIGURE P6-35 Units: Inches NOTE: Two steel pipes connected by 0.50 in plate

Part Area y Ay IcdAd

2

Ic + Ad2

1-Vert plate* 2.050 0.000 0.000 2.8717 0.000 0.000 2.872

FIGURE P6-36 Units: Inches NOTE: Four steel pipes connected as a rectangle

Part Area y Ay IcdAd

2

Ic + Ad2

1-Top pipes (2) 4.4560 12.0000 53.4720 6.0340 12.00 641.66 647.70

FIGURE P6-37 Units: Inches NOTE: Fabricated box shape; two angles + top and bottom plate

Part Area y Ay IcdAd

2

Ic + Ad2

1-Base plate 2.500 0.125 0.313 0.0130 2.592 16.800 16.813

FIGURE P6-38 Units: mm NOTE: Two SI Channels from App. A–6(d), mated with verticle webs

Part Area y Ay IcdAd

2

Ic + Ad2

1-Channel 950.0 0.000 0.000 1.433E+06 0.000 0.000 1.433E+06

FIGURE P6-39 Units: Inches NOTE: Fabricated box shape with 2 angles+2 channels+2 plates

Part Area y Ay IcdAd

2

Ic + Ad2

1-Angles (2) 1.888 0.586 1.106 0.6920 2.023 7.730 8.422

FIGURE P6-40 Units: Inches NOTE: W6x15 steel shape; 2 angles on bottom flange

Part Area y Ay IcdAd

2

Ic + Ad2

1-W6×15 4.430 2.995 13.268 29.100 0.642 1.827 30.927

FIGURE P6-41 Units: mm NOTE: Two SI 100x50×4 rectangular tubes from App. A-8(c)

Part Area y Ay IcdAd

2

Ic + Ad2

FIGURE P6-42 Units: Inches NOTE: 6x2x1/4 steel tube with two steel channels top and bottom

Part Area y Ay IcdAd

2

Ic + Ad2

1-Bot channel 1.590 0.457 0.727 0.3120 2.904 13.411 13.723

2-6x2×1/4 3.370 3.184 10.730 13.1000 0.177 0.1058 13.2058

FIGURE P6-43 Units: Inches NOTE: Horizontl I-shape on inverted channel

Part Area y Ay IcdAd

2

Ic + Ad2

1-Bot channel 1.881 0.730 1.373 0.980 0.321 0.193 1.173

FIGURE P6-44 Units: Inches NOTE: W-beam with inverted channel on top

Part Area y Ay IcdAd

2

Ic + Ad2

1-W12x30 8.790 6.150 54.059 238.000 2.022 35.953 273.953

2

1-W4×13 3.830 0.000 0.000 11.3000 0.000 0.000 11.300

2-bot tube 2.440 -3.080 -7.515 1.4800 3.080 23.1468 24.6268

FIGURE P6-46 Units: mm NOTE: SI IPE I–shape from App. A-7(e) & two angles on top flange

Part Area y Ay IcdAd

2

Ic + Ad2

1-I-300x50 5381.0 150 8.072E+05 8.356E+07 41.6 9.297E+06 9.286E+07

FIGURE P6-47 Units: Inches NOTE: Rectangular tube with top and bottom plates added

Part Area y Ay IcdAd

2

Ic + Ad2

1-6x2×1/4 3.370 0.000 0.000 13.1000 0.000 0.000 13.100

Y = 0.00 Inches Reference axis is Centroid by inspection

FIGURE P6-48 Units: Inches NOTE: Three C8x4.147 aluminum channels fabricated together

Part Area y Ay IcdAd

2

Ic + Ad2

1-C8x4.147 3.526 4.000 14.104 37.40 1.023 3.692 41.092

FIGURE P6-49 Units: Inches NOTE: Two steel angles with two plates fabricated to tee shape

Part Area y Ay IcdAd

2

Ic + Ad2

1-1/2x18 plate 9.000 0.250 2.250 0.1875 1.857 31.036 31.223

FIGURE P6-50 Units: Inches NOTE: Two rectangular tubes with channel on top; legs up

Part Area y Ay IcdAd

2

Ic + Ad2

1-1.50x0.50 tube 0.422 0.750 0.317 0.0973 0.255 0.027 0.125

FIGURE P6-51 Units: Inches NOTE: Two wood I-shapes with top and bottom wood boards

Part Area y Ay IcdAd

2

Ic + Ad2

1-14x2.5x1.375 11.094 0.000 0.000 319.500 0.000 0.00 319.50

FIGURE P6-52 Units: Inches NOTE: Two Small I-shapes with top plate

Part Area y Ay IcdAd

2

Ic + Ad2

1-1.95x1.00x.05 0.193 0.975 0.188 0.1770 0.453 0.040 0.217

FIGURE P6-53 Units: Inches NOTE: Two small angles on bottom plate

Part Area y Ay IcdAd

2

Ic + Ad2

1-L2x1.25x.188 0.574 0.936 0.537 0.2320 0.459 0.121 0.353

FIGURE P6-54 Units: mm NOTE: Three small SI tubes on underside of channel; legs down

Part Area y Ay IcdAd

2

Ic + Ad2

1-C 125x80 2152.0 54.6 117499.2 1.360E+06 1.190 3045 1.363E+06

2-Tube 20×2.5 137.4 62.0 8518.8 5.369E+03 6.210 5299.45 1.067E+04

FIGURE P6-55 Units: mm NOTE: Two DN 40 steel pipes connected by 12 mm plate

Part Area y Ay IcdAd

2

Ic + Ad2

1-Vert plate* 1220.88 0.00 0 1.053E+06 0.000 0.000E+00 1.053E+06

FIGURE P6-56 Units: in

NOTE: Three small tubes connected in triangular arrangement

Part Area y Ay IcdAd

2

Ic + Ad2

1-Bot tube* 0.1090 0.00 0 0.002460 1.000 0.109000 0.111460

.

NOTE: SOLUTIONS FOR RADIUS OF GYRATION PROBLEMS 6-57 TO 6-74

INCLUDED WITH PROBLEMS 6-2 TO 6-29

Each problem calls for calculation of the radius of gyration

with respect to the x-axis of the shape

Prob. No. Fig. No. rx

6-57 P6-2 3.04 in

6-58 P6-3 2.77 in

6-59 P6-4 70.35 mm

6-60 P6-5 21.99 mm

6-61 P6-6 8.78 mm